Tính C 2021 1 + C 2021 2 + C 2021 3
Tính \(C_{2021}^1 + C_{2021}^2 + C_{2021}^3\).
Ta có \(C_{2021}^1 + C_{2021}^2 + C_{2021}^3 = \frac{{2021!}}{{1!.2020!}} + \frac{{2021!}}{{2!.2019!}} + \frac{{2021!}}{{3!.2018!}}\).
\( = 2021!.\left( {\frac{1}{{2020.2019.2018!}} + \frac{1}{{2.2019.2018!}} + \frac{1}{{6.2018!}}} \right)\).
\( = \frac{{2021.2020.2019.2018!}}{{2018!}}.\left( {\frac{1}{{2020.2019}} + \frac{1}{{2.2019}} + \frac{1}{6}} \right)\).
\( = 2021.2020.2019.\left[ {\frac{1}{{2019}}\left( {\frac{1}{{2020}} + \frac{1}{2}} \right) + \frac{1}{6}} \right]\).
\[ = 2021.2020.2019.\left( {\frac{1}{{2019}}.\frac{{1011}}{{2020}} + \frac{1}{6}} \right)\].
\[ = 2021.1011 + \frac{{2021.1010.2.3.673}}{6}\].
= 2021.1011 + 2021.1010.673.
= 2021.(1011 + 679 730).
= 2021.680 741.
= 1 375 777 561.